RMS values for current and voltage - full calculus derivation [AC circuit physics]

00:00 Simple average values of current and voltage are USELESS in AC circuits, because they are just zero! How can we find the average in a way that captures what we really mean by average current and average voltage in an AC circuit? This is where RMS or root-mean-square averaging come to the rescue, and we justify it with all the geometric arguments AND calculus that it takes for physics and engineering majors to really understand RMS values.

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00:50 Current and voltage in a general AC circuit: current and voltage are sinusoidal functions that are, in general, out of phase. By convention, we give the current function a phase angle of zero, so i(t)=Icos(omega*t) and the voltage function has a phase angle of phi, so v(t)=Vcos(omega*t+phi).

02:24 Calculus review: the average value of a function. We develop a formula for the average value of a function starting from the geometric meaning: the constant function bounding the same area. Using the geometry, we get a definition of average value as 1/(b-a)*integral_a^b f(x)dx.

03:59 Simple average value of a periodic function: to take the average of a periodic function, we only need to take the average over one period. In the large time limit, any partial period covered at the end of the interval only causes a negligible amount of error, so we can just average over one period (or any integer number of periods) to obtain the average value!

07:52 Simple average value of sinusoidal current and voltage functions: we find the average value of the current function and voltage function and - no shock here - these averages are zero! We include the details of the average value integrals though!

10:12 RMS: a better way to average. To circumvent the problem of sinusoidal functions having an average value of zero, we start by squaring the function in order to force the values to be positive. We note that the square of a sinusoidal function has half the period of the original function, but we take the average value of the square over the original integration interval anyway (two periods). Finally, we're going to square root the mean of the square of the function, giving us back the original units for current or voltage, and this is the RMS average!

12:53 Relating RMS to amplitude: we compute the RMS integrals for current and voltage and show that i_rms=I/sqrt(2) and v_rms=V/sqrt(2). The voltage integral requires an interesting periodicity argument!

17:34 Simple example: given the rms voltage at a wall outlet, we compute the peak voltage, or voltage amplitude.

Part of my AC circuit physics miniseries:

Part 1: the current phasor and the voltage phasor for a resistor:
👉    • The current phasor and resistor voltage ph...  

Part 2: the voltage phasor for an inductor:
👉    • The inductor voltage phasor and inductive ...  

Part 3: the voltage phasor for a capacitor:
👉    • The capacitor voltage phasor and capacitiv...  

Part 4: phasor analysis of the RLC circuit:
👉    • RLC series circuit phasor analysis, impeda...  

Part 5: RMS values for current and voltage - full calculus derivations.
👉    • RMS values for current and voltage - full ...  

Part 6: Power calculations for inductors, capacitors and resistors.
👉    • Power in AC circuit elements - full calcul...  

Part 7: Power and the power factor in the RLC series circuit.
👉    • Power and the power factor in the series R...  

Part 8: Resonance in the RLC circuit + RLC power calculations. [TBA]

#accircuits #RMS #physics